Effects of Lateral Surface Conditions in Time-Harmonic Nonsymmetric Wave Propagation in a Cylinder

1989 ◽  
Vol 56 (4) ◽  
pp. 910-917 ◽  
Author(s):  
Yoon Young Kim ◽  
Charles R. Steele
Keyword(s):  
Lateral Surface ◽  
Present Method ◽  
Lateral Wall ◽  
Computational Effort ◽  
Wave Solutions ◽  
Time Harmonic ◽  
End Conditions ◽  
Analogous Manner ◽  
Method Of Solution ◽  
Free Case

The present work is a part of the effort toward the development of an efficient method of solution to handle general nonsymmetric time-harmonic end conditions in a cylinder with a traction-free lateral surface. Previously, Kim and Steele (1989a) develop an approach for the general axisymmetric case, which utilizes the well-known uncoupled wave solutions for a mixed lateral wall condition. For the case of a traction-free lateral wall, the uncoupled wave solutions provide: (1) a convenient set of basis functions and (2) approximations for the relation between end stress and displacement which are asymptotically valid for high mode index numbers. The decay rate with the distance from the end is, however, highly dependent on the lateral wall conditions. The present objective was to demonstrate that the uncoupled solutions of the nonsymmetric waves discussed by Kim (1989), which satisfy certain mixed lateral wall conditions, can be utilized in an analogous manner for the asymptotic analysis of the traction-free case. Results for the end displacement/stress due to various end conditions, computed by the present method and by a more standard collocation method, were compared. The present method was found to reduce the computational effort by orders of magnitude.

Gaussian wave packets in inhomogeneous media with curved interfaces

1987 ◽  
Vol 412 (1842) ◽  
pp. 93-123 ◽  
Keyword(s):  
Wave Equation ◽  
Large Scale ◽  
Evolution Equations ◽  
Wave Packets ◽  
Present Theory ◽  
Orthogonal Direction ◽  
Time Harmonic ◽  
Method Of Solution ◽  
Classical Wave Equation ◽  
The Time Domain

Time-dependent particle-like pulses are considered as asymptotic solutions of the classical wave equation. The wave packets are localized in space with gaussian envelopes. The pulse centres propagate along the rays of the wave equation, and the envelope parameters satisfy evolution equations very similar to the ray equations for time-harmonic disturb­ances. However, the present theory contains an extra degree of freedom not found in the time-harmonic theory. Explicit results are presented for media with constant velocity gradients, and interesting new phenomena are identified. For example, a pulse that is initially long in the direction of propagation and comparatively narrow in the orthogonal direction, maintains its initial spatial orientation even as the propagation direction rotates. The reflection and transmission of a pulse incident upon an interface are also discussed. The various theoretical results are illustrated by numerical simulations. This method of solution could be very useful for fast forward modelling in large-scale structures. It is formulated explicitly in the time domain and does not suffer from unphysical singularities at caustics.

A new method of solution for one-dimensional quasi-neutral bounded plasmas

2010 ◽  
Vol 76 (3-4) ◽  
pp. 617-625 ◽  
Author(s):  
M. KAMRAN ◽  
S. KUHN
Keyword(s):  
Present Method ◽  
Analytic Solution ◽  
Ion Source ◽  
New Method ◽  
Ion Density ◽  
One Dimensional ◽  
Plasma Equation ◽  
Neutrality Condition ◽  
Bounded Plasma ◽  
Method Of Solution

AbstractA new method is proposed for calculating the potential distribution Φ(z) in a one-dimensional quasi-neutral bounded plasma; Φ(z) is assumed to satisfy a quasi-neutrality condition (plasma equation) of the form ni{Φ(z)} = ne(Φ), where the electron density ne is a given function of Φ and the ion density ni is expressed in terms of trajectory integrals of the ion kinetic equation. While previous methods relied on formally solving a global integral equation (Riemann, Phys. Plasmas, vol. 13, 2006, paper no. 013503; Kos et al., Phys. Plasmas, vol. 16, 2009, paper no. 093503), the present method is characterized by piecewise analytic solution of the plasma equation in reasonably small intervals of z. As a first concrete application, Φ(z) is found analytically through order z4 near the center of a collisionless Tonks–Langmuir discharge with a cold-ion source.

Flexural Wave Solutions of Coupled Equations Representing the More Exact Theory of Bending

1953 ◽  
Vol 20 (4) ◽  
pp. 511-514
Author(s):  
Julius Miklowitz
Keyword(s):  
Boundary Conditions ◽  
Present Method ◽  
Shear Force ◽  
Beam Element ◽  
Transverse Load ◽  
Flexural Wave ◽  
Infinite Beam ◽  
Coupled Equations ◽  
Wave Solutions ◽  
The Moment

Abstract Presented here is a new method for deriving flexural wave solutions for the Timoshenko bending theory. The method is based on a breakdown of the total deflection into its bending and shear components. Instead of treating the full Timoshenko equation (1) an equivalent set of coupled equations, representing the rotational and translatory motions of the beam element, is solved. The advantages of this method stem from (a) the simplicity of the associated expressions for the moment and shear force, which are the elementary bending theory relations, and (b) the well-defined nature of the related boundary conditions. The latter is particularly important since it is difficult to define the proper boundary conditions associated with the full Timoshenko equation. This is evidenced in the works of Uflyand (2) and Dengler and Goland (3), both of which are concerned with wave solutions for the infinite beam under the action of a concentrated transverse load. The quoted work (3) points out the erroneous boundary conditions used in the Uflyand work (2). The present method is applied to the same case treated in the works (2, 3). Agreement is shown with the Dengler and Goland solution. The Uflyand solution is shown to have meaning when interpreted properly. The derivation of transforms for other beam cases, both finite and infinite, by the present method has also been included in this work.

Analysis of Tapered Laminated Composites with Non-Symmetric Lay-Up

1997 ◽  
Vol 16 (7) ◽  
pp. 631-660 ◽  
Author(s):  
B. Varughese ◽  
A. Mukherjee
Keyword(s):  
Stiffness Matrix ◽  
Present Method ◽  
Laminated Composites ◽  
Global Analysis ◽  
Computational Effort ◽  
Local Analysis ◽  
Local Approach ◽  
Fine Mesh ◽  
Conventional Analysis ◽  
Analysis Models

A global-local approach for the analysis of tapered laminated composites is presented. The method is considerably more economical than existing techniques. A new drop-off element that is degenerated from a 2-D drop-off element is introduced to carry out the global analysis. The drop-off element accommodates the termination of plies within the element and therefore, the size of the stiffness matrix reduces. At the vicinity of the drop-off, where the stress concentration is high, a local analysis with a refined mesh is performed with the results of the global analysis. Since a fine mesh is to be adopted only near the drop-off, there is substantial economy in the computational effort and time as compared to the conventional analysis models. The present method has been validated extensively against published results. The discussions on the stress/strain distributions at different locations have been presented.

Gridding with continuous curvature splines in tension

Geophysics ◽  
1990 ◽  
Vol 55 (3) ◽  
pp. 293-305 ◽  
Author(s):  
W. H. F. Smith ◽  
P. Wessel
Keyword(s):  
Elastic Plate ◽  
General System ◽  
Computational Effort ◽  
Tension Parameter ◽  
Plate Flexure ◽  
Inflection Points ◽  
Method Of Solution ◽  
Total Squared Curvature ◽  
Pass Through ◽  
Minimum Curvature

A gridding method commonly called minimum curvature is widely used in the earth sciences. The method interpolates the data to be gridded with a surface having continuous second derivatives and minimal total squared curvature. The minimum‐curvature surface has an analogy in elastic plate flexure and approximates the shape adopted by a thin plate flexed to pass through the data points. Minimum‐curvature surfaces may have large oscillations and extraneous inflection points which make them unsuitable for gridding in many of the applications where they are commonly used. These extraneous inflection points can be eliminated by adding tension to the elastic‐plate flexure equation. It is straightforward to generalize minimum‐curvature gridding algorithms to include a tension parameter; the same system of equations must be solved in either case and only the relative weights of the coefficients change. Therefore, solutions under tension require no more computational effort than minimum‐curvature solutions, and any algorithm which can solve the minimum‐curvature equations can solve the more general system. We give common geologic examples where minimum‐curvature gridding produces erroneous results but gridding with tension yields a good solution. We also outline how to improve the convergence of an iterative method of solution for the gridding equations.

Stress Concentration Due to a Hemispherical Pit at a Free Surface

1954 ◽  
Vol 21 (1) ◽  
pp. 57-62
Author(s):  
R. A. Eubanks
Keyword(s):  
Free Surface ◽  
Present Method ◽  
Infinite Medium ◽  
The Body ◽  
Hydrostatic Tension ◽  
Rotationally Symmetric ◽  
Method Of Solution ◽  
In Series ◽  
Axis Of Symmetry ◽  
Plane At Infinity

Abstract This paper contains a solution in series form for the stresses and displacements around a hemispherical pit at a free surface of an elastic body. The problem is idealized by considering a semi-infinite medium which otherwise is bounded by a plane. At infinity the body is assumed to be in a state of plane hydrostatic tension perpendicular to the axis of symmetry of the pit. The present method of solution may be generalized to loadings which are not rotationally symmetric. Numerical results are given for the variation along the axis of symmetry of the normal stress which is parallel to the tractions at infinity; these results are compared with the known corresponding numerical values appropriate to the two-dimensional analog of the present problem.

A Kind of Optimization Method on Plate-Shell Structures with Stiffeners

2015 ◽  
Vol 723 ◽  
pp. 234-239
Author(s):  
Jian Kang Li ◽  
Kai Ma
Keyword(s):  
Strain Energy Density ◽  
Present Method ◽  
Strain Energy ◽  
Structural Parameters ◽  
Optimization Method ◽  
Layout Optimization ◽  
Computational Effort ◽  
Parameters Optimization ◽  
Shell Structures ◽  
Powell Method

An new optimization method which combines layout optimization of stiffeners with structural parameters optimization of structures is discussed. The first step is to optimize the layout of stiffenerser, then the structural parameters were optimized. In layout optimization of stiffeners, the strain energy density sensitivities of the elements were used to determine which elements of stiffeners should be deleted. In structural parameters optimization, the object function and the constraint functions were approximated by the second-order Taylor expansion. DFP (Davidon, Fletcher and Powell) method was presented to solve optimal problem. In order to reduce computational effort, the combined approximation (CA) method was used to reanalysis and update the displacements and stresses of the structure. The present method was applied to a bunker. The numerical results show that the optimization method was effective for optimization of plate-shell structure with stiffeners, and it could be to implement on a computer.

An Improved Technique for Elastodynamic Green's Function Computation for Transversely Isotropic Solids

2019 ◽  
Vol 2 (2) ◽  
Author(s):  
Samaneh Fooladi ◽  
Tribikram Kundu
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Composite Structures ◽  
Transversely Isotropic ◽  
Computational Effort ◽  
Computational Time ◽  
Function Computation ◽  
Time Harmonic ◽  
Transversely Isotropic Solids ◽  
Isotropic Solids

Elastodynamic Green's function for anisotropic solids is required for wave propagation modeling in composites. Such modeling is needed for the interpretation of experimental results generated by ultrasonic excitation or mechanical vibration-based nondestructive evaluation tests of composite structures. For isotropic materials, the elastodynamic Green’s function can be obtained analytically. However, for anisotropic solids, numerical integration is required for the elastodynamic Green's function computation. It can be expressed as a summation of two integrals—a singular integral and a nonsingular (or regular) integral. The regular integral over the surface of a unit hemisphere needs to be evaluated numerically and is responsible for the majority of the computational time for the elastodynamic Green's function calculation. In this paper, it is shown that for transversely isotropic solids, which form a major portion of anisotropic materials, the integration domain of the regular part of the elastodynamic time-harmonic Green's function can be reduced from a hemisphere to a quarter-sphere. The analysis is performed in the frequency domain by considering time-harmonic Green's function. This improvement is then applied to a numerical example where it is shown that it nearly halves the computational time. This reduction in computational effort is important for a boundary element method and a distributed point source method whose computational efficiencies heavily depend on Green's function computational time.

Modifications of Series Expansions for General End Conditions and Corner Singularities on the Semi-Infinite Strip

1990 ◽  
Vol 57 (3) ◽  
pp. 581-588 ◽  
Author(s):  
Yoon Young Kim ◽  
Charles R. Steele
Keyword(s):  
Stiffness Matrix ◽  
Harmonic Wave ◽  
Infinite Cylinder ◽  
Series Expansions ◽  
Infinite Strip ◽  
Corner Singularities ◽  
Numerical Examples ◽  
Time Harmonic ◽  
End Conditions ◽  
Expansion Coefficients

Modified series expansions are used to study semi-infinite isotropic elastic strip problems for general end conditions and corner singularities. The solutions of strips with mixed lateral edges are used as the expansion sets of the end displacement and stress, and an end stiffness matrix, the relation of harmonics of the end displacement and stress, is formed. The present end stiffness matrix approach, an extension to static strip problems of the method by Kim and Steele (1989, 1990) for time-harmonic wave propagation in a semi-infinite cylinder, is effective due to the asymptotic behavior of the stiffness matrix. Also presented is a technique for handling the corner singularities, which is based on the asymptotic analysis of the expansion coefficients of the end stresses. With this, the order and strength of the singularities are determined, local oscillations are virtually suppressed, and converging solutions are obtained. Some numerical examples are given to demonstrate the effectiveness of the approach.

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